Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

In this paper the nonlinear wave equation $$u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0$$ is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.     

Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ^d and translation-invariant Gibbs states for a related Hamiltonian on ( ^d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.     

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation

Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

Ground state of a spin-phonon system. I. Variational estimates

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve (B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and whenB>e ^–1 and an infinite-order transition whenB<e ^–1, but is trivial in the large- region. The classical theory yields discontinuities in the second derivatives but is trivial in the small- region. An improved variationally fixed ground-state wave function is analyzed. It gives a new (B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²–1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire (B) plane. The crossover to classical behavior occurs near =1/2 forB1. But the wave function does not describe quantum fluctuations in the large- phase. A second way of combining source and classical effects is described. It yields a second-order transition (near =1/2 forB1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.     

Structural mining: self-consistent design on flexible protein-peptide docking and transferable binding affinity potential

A flexible protein-peptide docking method has been designed to consider not only ligand flexibility but also the flexibility of the protein. The method is based on a Monte Carlo annealing process. Simulations with a distance root-mean-square (dRMS) virtual energy function revealed that the flexibility of protein side chains was as important as ligand flexibility for successful protein-peptide docking. On the basis of mean field theory, a transferable potential was designed to evaluate distance-dependent protein-ligand interactions and atomic solvation energies. The potential parameters were developed using a self-consistent process based on only 10 known complex structures. The effectiveness of each intermediate potential was judged on the basis of a Z score, approximating the gap between the energy of the native complex and the average energy of a decoy set. The Z score was determined using experimentally determined native structures and decoys generated by docking with the intermediate potentials. Using 6600 generated decoys and the Z score optimization criterion proposed in this work, the developed potential yielded an acceptable correlation of R(2) = 0.77, with binding free energies determined for known MHC I complexes (Class I Major Histocompatibility protein HLA-A(*)0201) which were not present in the training set. Test docking on 25 complexes further revealed a significant correlation between energy and dRMS, important for identifying native-like conformations. The near-native structures always belonged to one of the conformational classes with lower predicted binding energy. The lowest energy docked conformations are generally associated with near-native conformations, less than 3.0 Angstrom dRMS (and in many cases less than 1.0 Angstrom) from the experimentally determined structures.     

The separation of palladium and platinum by ion flotation

Differences in the ion flotation properties of palladium(II) and platinum(IV) chloro complexes in aqueous solutions are used to achieve separations of these metals. The anionic chloro complex PtCl^2-₆ is floated selectively with cationic surfactants of the type, RNR'₃Br, from solutions of PdCl^2-₄ and various concentrations of hydrochloric acid. The palladium(II) does not float from solutions of ? 3.0 M HCl and the platinum(IV) floated from these solutions can be recovered free of palladium. However, the separation is incomplete as much of the platinum(IV) is also unfloated from these solutions. Quantitative separations are obtained by conversion of the palladium(II) to the cationic ammine, Pd(NH₃)₄²⁺ with aqueous ammonia prior to flotation. The anionic chloro complex of platinum(IV) is unaffected by the presence of ammonia and is floated quantitatively with the surfactant n-hexadecyltri-n-propylammonium bromide from 0.01 M ammonia solutions.     

Surfactant displacement of human serum albumin adsorbed on loosely packed self-assembled monolayers: cetyltrimethylammonium bromide versus sodium dodecyl sulfate

The surfactants sodium dodecyl sulfate (SDS) and cetyltrimethylammonium bromide (CTAB) displace human serum albumin (HSA) from loosely packed self-assembled monolayers (SAM) of hydrophobic alkyl chains by different means. Removal of HSA is of interest because previous work has suggested that the adsorption of HSA to such loosely packed SAMs may be sufficiently tenacious to offer opportunities for surface passivation. While HSA remains on the surface after exposure to SDS and rinsing, no protein remains after exposure to CTAB and rinsing. X-ray reflectivity and X-ray photoelectron spectroscopy measurements indicate that CTAB molecules remain interdigitated in the loosely packed SAM after rinsing, suggesting that CTAB is more effective in removing the HSA because it interacts more strongly with the SAM.     

Probing metabolic alterations in breast cancer in response to molecular inhibitors with Raman spectroscopy and validated with mass spectrometry

Rapid and accurate response to targeted therapies is critical to differentiate tumors that are resistant to treatment early in the regimen. In this work, we demonstrate a rapid, noninvasive, and label-free approach to evaluate treatment response to molecular inhibitors in breast cancer (BC) cells with Raman spectroscopy (RS). Metabolic reprogramming in BC was probed with RS and multivariate analysis was applied to classify the cells into responsive or nonresponsive groups as a function of drug dosage, drug type, and cell type. Metabolites identified with RS were then validated with mass spectrometry (MS). We treated triple-negative BC cells with Trametinib, an inhibitor of the extracellular-signal-regulated kinase (ERK) pathway. Changes measured with both RS and MS corresponding to membrane phospholipids, amino acids, lipids and fatty acids indicated that these BC cells were responsive to treatment. Comparatively, minimal metabolic changes were observed post-treatment with Alpelisib, an inhibitor of the mammalian target of rapamycin (mTOR) pathway, indicating treatment resistance. These findings were corroborated with cell viability assay and immunoblotting. We also showed estrogen receptor-positive MCF-7 cells were nonresponsive to Trametinib with minimal metabolic and viability changes. Our findings support that oncometabolites identified with RS will ultimately enable rapid drug screening in patients ensuring patients receive the most effective treatment at the earliest time point.

Rapid and accurate response to targeted therapies is critical to differentiate tumors that are resistant to treatment early in the regimen.